We find that the classical composers would often add repeats to parts of their music. In playing the part a second time, we see that the piece is not just a musing, an improvisation; it underscores the intentionality of the piece.
When an accomplished dancer raises one hand asymmetrically, it is an asymmetry based on movement of a symmetric body that wants to accomplish moving and still remain itself, symmetrical. Hence, other parts of the body compensate to balance the asymmetry, placing the symmetry in the progression, the movement, the dance.
The same applies to drawing, as I have learned from the great Disney animator. Will Shortz - when a character moves part of his body in one direction, another part compensates in the other.
Of course, of symmetry, we are not just talking about a horizontal mirroring but an integrity of the piece that while it bends to function (or to movement, as in a dance), it remains.
Yes, there many asymmetric exceptions of beauty like the Bilbao Guggenheim, baroque pearls, and Sammy Davis' mismatched eyes. These are the artistry of novelty, that is their beauty. In the Guggenheim, there is the integrity of one simple beautiful color, the symmetry with regard to it level base, and to it's internal design and function. And Sammy's beauty was in his voice and his otherwise symmetric face and wonderful jawline. As to baroque pearls, it is their surrender to the more powerful environment while retaining their Bilbao like luster that makes them treasures.